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# 56

Please do not rely on any information it contains.

56 is an integer, the number of reduced 5 by 5 Latin squares (see A000315).

## Membership in core sequences

 Even numbers ..., 50, 52, 54, 56, 58, 60, 62 ... A005843 Composite numbers ..., 52, 54, 55, 56, 57, 58, 60, ... A002808 Abundant numbers ..., 42, 48, 54, 56, 60, 66, 70, ... A005101 Partition numbers ..., 22, 30, 42, 56, 77, 101, ... A000041 Tetrahedral numbers 1, 4, 10, 20, 35, 56, 84, 120, ... A000292 Oblong numbers ..., 20, 30, 42, 56, 72, 90, 110, ... A002378 Quarter-squares ..., 36, 42, 49, 56, 64, 72, 81, ... A002620 Central polygonal numbers ..., 29, 37, 46, 56, 67, 79, 92, ... A000124

In Pascal's triangle, 56 occurs four times, the first two times on row 8 as the sum of 21 and 35 from row 7.

Notice 56 as a denominator in this continued fraction (A002530):

${\displaystyle {\sqrt {3}}=1+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{4+{\cfrac {1}{11+{\cfrac {1}{15+{\cfrac {1}{41+{\cfrac {1}{56+{\cfrac {1}{153+\ddots }}}}}}}}}}}}}}}}.}$

Also ${\displaystyle {\frac {41}{56}}\approx -1+{\sqrt {3}}}$.

## Sequences pertaining to 56

 Divisors of 56 1, 2, 4, 7, 8, 14, 28, 56 A018265 Multiples of 56 0, 56, 112, 168, 224, 280, 336, 392, 448, 504, 560, 616, 672, ... ${\displaystyle 3x-1}$ sequence starting at 36 36, 18, 9, 26, 13, 38, 19, 56, 28, 14, 7, 20, 10, 5, 14, 7, 20, ... A008894 ${\displaystyle 3x+1}$ sequence starting at 99 99, 298, 149, 448, 224, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, ... A008882 ${\displaystyle 5x+1}$ sequence starting at 5 5, 26, 13, 66, 33, 11, 56, 28, 14, 7, 36, 18, 9, 3, 1, 6, 3, 1, ... A057688

## Partitions of 56

There are 526823 partitions of 56.

The Goldbach representations of 56 are: 3 + 53 = 13 + 43 = 19 + 37.

## Roots and powers of 56

In the table below, irrational numbers are given truncated to eight decimal places.

TABLE GOES HERE

## Values for number theoretic functions with 56 as an argument

 ${\displaystyle \mu (56)}$ 0 ${\displaystyle M(56)}$ −2 ${\displaystyle \pi (56)}$ 16 ${\displaystyle \sigma _{1}(56)}$ 120 ${\displaystyle \sigma _{0}(56)}$ 8 ${\displaystyle \phi (56)}$ 24 ${\displaystyle \Omega (56)}$ 4 ${\displaystyle \omega (56)}$ 2 ${\displaystyle \lambda (56)}$ 6 This is the Carmichael lambda function. ${\displaystyle \lambda (56)}$ 1 This is the Liouville lambda function.

## Factorization of 56 in some quadratic integer rings

As was mentioned above, 56 is composite in ${\displaystyle \mathbb {Z} }$. But it has different factorizations in some quadratic integer rings.

TABLE GOES HERE

## Representation of 56 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 111000 2002 320 211 132 110 70 62 56 51 48 44 40 3B 38 35 32 2I 2G

 ${\displaystyle -1}$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1729